unique intersection between k lines ($k>2$)

Question

I am looking for a closed form solution for the following problem:

Given $k>2$ lines with a joint intersection $(\hat{x},\hat{y})$,

and $n$ noisy measurements from each line (white gaussian noise): $$(x_{i1},y_{k1}),(x_{i1},y_{i2}),...,(x_{in},y_{in}) \ \ ,\ \ \ 1\leq i \leq K $$

The goal - estimate the intersection point. (the actual lines params are not known, the intersection point is not known, only the measurements are known)

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a 'naive' method:

1. find the least-squares solution for each line (separately)
2. find the "intersection", as the point that minimizes the $l_2$ distance to each line

But, I am looking for a closed form solution that directly finds the intersection, meaning a least squares / iterative process.

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Ideas:

1. parametric function $(x, y) = (\hat{x},\hat{y}) + t \cdot v$,$\ \ $ where $v=(m_x,m_y)$
2. point-slope representation: $(y-\hat{y})-m(x-\hat{x})=0$

Thank you for any suggestions